Stationary Wong–Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations

Abstract

In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions.